Linear representation theory of semidihedral group:SD16
This article gives specific information, namely, linear representation theory, about a particular group, namely: semidihedral group:SD16.
View linear representation theory of particular groups | View other specific information about semidihedral group:SD16
Summary
We shall use the semidihedral group of order 16 with the following presentation:
.
Item | Value |
---|---|
degrees of irreducible representations over a splitting field (e.g., ![]() ![]() |
1,1,1,1,2,2,2 (1 occurs 4 times, 2 occurs 3 times) maximum: 2, lcm: 2, number: 7, sum of squares: 16 |
Schur index values of irreducible representations over a splitting field | 1,1,1,1,1,1,1 |
smallest ring of realization (characteristic zero) | ![]() ![]() ![]() Same as ring generated by character values. |
minimal splitting field, i.e., smallest field of realization (characteristic zero) | ![]() ![]() ![]() Same as field generated by character values, because all Schur index values are 1. See minimal splitting field need not be cyclotomic |
condition for a field to be a splitting field | The characteristic should not be equal to 2, and the polynomial ![]() For a finite field of size ![]() ![]() ![]() |
minimal splitting field (characteristic ![]() |
Case ![]() ![]() ![]() Case ![]() ![]() ![]() |
smallest size splitting field | Field:F3. |
degrees of irreducible representations over the rational numbers | 1,1,1,1,2,4 |
Representations
Summary information
Below is summary information on irreducible representations that are absolutely irreducible, i.e., they remain irreducible in any bigger field, and in particular are irreducible in a splitting field. We assume that the characteristic of the field is not 2, except in the last column, where we consider what happens in characteristic 2.
Name of representation type | Number of representations of this type | Degree | Schur index | Criterion for field | Kernel (a normal subgroup of semidihedral group:SD16 -- see subgroup structure of semidihedral group:SD16) | Quotient by kernel (on which it descends to a faithful representation) | Characteristic 2 |
---|---|---|---|---|---|---|---|
trivial | 1 | 1 | 1 | any | whole group | trivial group | works |
sign representation with kernel ![]() |
1 | 1 | 1 | any | Z8 in SD16: ![]() |
cyclic group:Z2 | works, same as trivial |
sign representation with kernel a maximal dihedral subgroup | 1 | 1 | 1 | any | D8 in SD16: ![]() |
cyclic group:Z2 | works, same as trivial |
sign representation with kernel a maximal quaternion subgroup | 1 | 1 | 1 | any | Q8 in SD16: ![]() |
cyclic group:Z2 | works, same as trivial |
two-dimensional irreducible, not faithful | 1 | 2 | 1 | any | center of semidihedral group:SD16: ![]() |
dihedral group:D8 | indecomposable but not irreducible |
two-dimensional faithful irreducible | 2 | 2 | 1 | The polynomial ![]() ![]() |
trivial subgroup, i.e., it is a faithful linear representation | semidihedral group:SD16 | ? |
Below are representations that are irreducible over a non-splitting field, but split over a splitting field.
Name of representation type | Number of representations of this type | Degree | Criterion for field | What happens over a splitting field? | Kernel | Quotient by kernel (on which it descends to a faithful representation) |
---|---|---|---|---|---|---|
four-dimensional faithful irreducible | 1 | 4 | The polynomial ![]() ![]() |
splits into the two two-dimensional faithful irreducibles. | trivial subgroup, i.e., it is a faithful linear representation | semidihedral group:SD16 |
Trivial representation
The trivial representation or principal representation (whose character is called the trivial character or principal character) sends all elements of the group to the matrix
:
Element | Matrix | Characteristic polynomial | Minimal polynomial | Trace, character value |
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1 |
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1 |
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1 |
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1 |
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1 |
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1 |
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1 |
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1 |
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1 |
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1 |
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1 |
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1 |
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1 |
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1 |
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1 |
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1 |
Sign representation with kernel 
This representation is a one-dimensional representation sending everything in the cyclic subgroup (see Z8 in SD16) to
and everything outside it to
.
To keep the description short, we club together the cosets rather than having one row per element:
Elements | Matrix | Characteristic polynomial | Minimal polynomial | Trace, character value |
---|---|---|---|---|
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1 |
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-1 |
Sign representation with kernel 
There is a sign representation with kernel which is dihedral group:D8 (see D8 in SD16). Everything inside the subgroup goes to
and everything outside the subgroup goes to
.
To keep the descriptions short, we club together the cosets rather than having one row per element:
Elements | Matrix | Characteristic polynomial | Minimal polynomial | Trace, character value |
---|---|---|---|---|
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1 |
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-1 |
Sign representation with kernel 
There is a sign representation with kernel which is quaternion group (see Q8 in SD16). Everything inside the subgroup goes to
and everything outside the subgroup goes to
.
To keep the descriptions short, we club together the cosets rather than having one row per element:
Elements | Matrix | Characteristic polynomial | Minimal polynomial | Trace, character value |
---|---|---|---|---|
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1 |
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-1 |
Two-dimensional irreducible unfaithful representation
This representation has kernel equal to -- center of semidihedral group:SD16. It descends to a faithful irreducible two-dimensional representation of the quotient group, which is isomorphic to dihedral group:D8.
To keep the descriptions short, we club together the cosets rather than having one row per element:
Element | Matrix | Characteristic polynomial | Minimal polynomial | Trace, character value | Determinant |
---|---|---|---|---|---|
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2 | 1 |
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0 | 1 |
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-2 | 1 |
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0 | 1 |
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0 | -1 |
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0 | -1 |
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0 | -1 |
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0 | -1 |
Two-dimensional faithful irreducible representations
Four-dimensional faithful irreducible representation over a non-splitting field
Character table
FACTS TO CHECK AGAINST (for characters of irreducible linear representations over a splitting field):
Orthogonality relations: Character orthogonality theorem | Column orthogonality theorem
Separation results (basically says rows independent, columns independent): Splitting implies characters form a basis for space of class functions|Character determines representation in characteristic zero
Numerical facts: Characters are cyclotomic integers | Size-degree-weighted characters are algebraic integers
Character value facts: Irreducible character of degree greater than one takes value zero on some conjugacy class| Conjugacy class of more than average size has character value zero for some irreducible character | Zero-or-scalar lemma
Below is the character table over a splitting field, where stands for any chosen square root of
:
Representation/conjugacy class representative | ![]() |
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1 | 1 | 1 | 1 | 1 | -1 | -1 |
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1 | 1 | 1 | -1 | -1 | 1 | -1 |
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1 | 1 | 1 | -1 | -1 | -1 | 1 |
two-dimensional unfaithful, kernel is center | 2 | 2 | -2 | 0 | 0 | 0 | 0 |
first faithful irreducible representation | 2 | -2 | 0 | ![]() |
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0 | 0 |
second faithful irreducible representation | 2 | -2 | 0 | ![]() |
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0 | 0 |